Dehn filling and the knot group I: Realization Property

Abstract

Each $r$-Dehn filling of the exterior $E(K)$ of a knot $K$ in $S^3$ produces a $3$-manifold $K(r)$, and induces an epimorphism from the knot group $G(K) = \pi_1(E(K))$ to $\pi_1(K(r))$, which trivializes elements in its kernel. To each element $g \in G(K)$, consider all the non-trivial Dehn fillings and assign $\mathcal{S}_K(g) = \{ r \in \mathbb{Q} \mid \textrm{$r$-Dehn filling trivializes}\ g \}$ $\subset \mathbb{Q}$. Which subsets of $\mathbb{Q}$ can occur as $\mathcal{S}_K(g)$? Property P concerns this question and gives a fundamental result which asserts that the emptyset can be realized by $\mathcal{S}_K(\mu)$ for the meridian $\mu$ of $K$. Suppose that $K$ is a hyperbolic knot. Then $\mathcal{S}_K(g)$ is known to be finite for all non-trivial elements $g \in G(K)$. We prove that generically, for instance, if $K$ has no exceptional surgery, then any finite (possibly empty) family of slopes $\mathcal{R} = \{ r_1, . . . , r_n \}$ can be realized by $\mathcal{S}_K(g)$ for some element $g \in G(K)$. Furthermore, there are infinitely many, mutually non-conjugate such elements, each of which is not conjugate to any power of $g$. We also provide an example showing that the above realization property does not hold unconditionally.